Classification and nondegeneracy of SU(n+1) Toda system with singular sources

Changshou Lin (1) , Juncheng Wei (2) , Dong Ye (3)

1 NTU - National Taiwan University [Taiwan]
2 CUHK - The Chinese University of Hong Kong [Hong Kong]
3 LMAM - Laboratoire de Mathématiques et Applications de Metz

We consider the following Toda system\begin{align*}\Delta u_i + \displaystyle \sum_{j = 1}^n a_{ij}e^{u_j} = 4\pi\gamma_{i}\delta_{0} \;\; \text{in }\mathbb R^2, \quad\displaystyle \int_{\mathbb R^2}e^{u_i} dx < \infty,\;\; \forall\; 1\leq i \leq n,\end{align*}where $\gamma_{i} > -1$, $\delta _0$ is Dirac measure at $0$,and the coefficients $a_{ij}$ form the standard tri-diagonalCartan matrix.In this paper, (i)we completely classify the solutions and obtain the quantization result:$$\sum_{j=1}^n a_{ij}\int _{\mathbb R^2}e^{u_j} dx = 4\pi (2+\gamma_i+\gamma _{n+1-i}), \;\;\forall\; 1\leq i \leq n.$$This generalizes the classification result by Jost and Wang for $\gamma_i=0$, $\forall \;1\leq i\leq n$.(ii) We prove that if $\gamma_i+\gamma_{i+1}+\ldots+\gamma_j \notin \mathbb Z$ for all $1\leq i\leq j\leq n$, then any solution $u_i$ is \textit{radially symmetric} w.r.t.~$0$.(iii) We prove that the linearized equation at any solution is \textit{non-degenerate}.These are fundamental results in order to understand the bubbling behaviour of the Toda system.

Domaines: Equations aux dérivées partielles [math.AP]

Changshou Lin, Juncheng Wei, Dong Ye. Classification and nondegeneracy of SU(n+1) Toda system with singular sources. Inventiones Mathematicae, 2012, 190 (1), pp.169-207. ⟨10.1007/s00222-012-0378-3⟩. ⟨hal-01095018⟩